Basic Stress Equations - Fairfield University
Dr. D. B. Wallace
Basic Stress Equations
Internal Reactions:
Centroid of Cross Section
6 Maximum
y
(3 Force Components
& 3 Moment Components)
"Cut Surface"
Centroid of Cross Section
Shear Forces ( )
y
x
"Cut Surface"
Bending Moments ()
x
Vx Vy
P
z
Normal Force ()
Force Components
Mx My
T
z
Torsional Moment or Torque ( )
Moment Components
Normal Force:
Centroid y "Cut Surface"
x
Pz Axial Force
Axial Stress
= P A
l Uniform over the entire cross section. l Axial force must go through centroid.
Shear Forces:
y "y" Shear Force
x
Vy
z
y "x" Shear Force
x
Vx
z
Cross Section V
Point of interest LINE perpendicular to V through point of interest
b = Length of LINE on the cross section
y
Aa
Aa = Area on one side of the LINE Centroid of entire cross section Centroid of area on one side of the LINE
I = Area moment of inertia of entire cross section about an axis pependicular to V.
b g = V Aa y Ib
y = distance between the two centroids
Note: The maximum shear stress for common cross sections are:
Cross Section:
Cross Section:
Rectangular:
max = 3 2 V A
Solid Circular:
max = 4 3 V A
I-Beam or H-Beam:
flange web max = V A web
Thin-walled tube:
max = 2 V A
Basic Stress Equations
Dr. D. B. Wallace
Torque or Torsional Moment:
Solid Circular or Tubular Cross Section:
y
"Cut Surface"
x
Tz Torque
= Tr J
max
=
16 T D3
for solid circular shafts
e j max
=
16 T Do Do4 - Di4
for hollow shafts
r = Distance from shaft axis to point of interest
R = Shaft Radius
D = Shaft Diameter
J = D4 = R4
e j 32
2
J = Do4 - Di4
32
for solid circular shafts for hollow shafts
Rectangular Cross Section:
y Centroid
"Cut Surface"
x
Tz Torque
2 1
Torsional Stress
Cross Section: b
Note:
a
a > b
Method 1:
b g e j max = 1 = T 3a +1.8 b a2 b2
ONLY applies to the center of the longest side
Method 2:
1,2
=
1,2
T a b2
a/b
1
2
1.0
.208
.208
1.5
.231
.269
2.0
.246
.309
Use the appropriate from the table
3.0
.267
.355
on the right to get the shear stress at
4.0
.282
.378
either position 1 or 2.
6.0
.299
.402
8.0
.307
.414
10.0 .313
.421
.333
----
Other Cross Sections: Treated in advanced courses.
2
Basic Stress Equations
Dr. D. B. Wallace
Bending Moment
"x" Bending Moment
y x
Mx
z
= Mx y and = My x
Ix
Iy
"y" Bending Moment
y
x
My
z
Moments of Inertia:
where: Mx and My are moments about indicated axes y and x are perpendicular from indicated axes
Ix and Iy are moments of inertia about indicated axes
b
c h
I = b h3 h is perpendicular to axis 12
Z = I = b h2 c6
D
c
R
I = D4 = R4
64
4
Z = I = D3 = R3
c 32
4
Parallel Axis Theorem:
new axis
d
Area, A I = I + A d2
centroid
I = Moment of inertia about new axis
I = Moment of inertia about the centroidal axis
A = Area of the region d = perpendicular distance between the two axes.
Maximum Bending Stress Equations:
max =
Mc I
=
M Z
b g max
=
32 M D3
Solid Circular
a f max
=
6M b h2
Rectangular
The section modulus, Z, can be found in many tables of properties of common cross sections (i.e., I-beams, channels, angle iron, etc.).
Bending Stress Equation Based on Known Radius of Curvature of Bend, .
The beam is assumed to be initially straight. The applied moment, M, causes the beam to assume a radius of curvature, .
Before: After:
= Ey
E = Modulus of elasticity of the beam material
M
M
y = Perpendicular distance from the centroidal axis to the
point of interest (same y as with bending of a
straight beam with Mx).
= radius of curvature to centroid of cross section
3
Basic Stress Equations
Dr. D. B. Wallace
Bending Moment in Curved Beam:
Geometry:
nonlinear stress
distribution
M
o centroidal
axis
e
co
ci
i
rn
neutral axis
ri
centroid
y
ro
r
zrn =
A dA
area
e = r - rn
Stresses:
Any Position:
b g = -M y e A rn + y
Inside (maximum magnitude):
i
=
M ci e A ri
A = cross sectional area
rn = radius to neutral axis r = radius to centroidal axis
e = eccentricity
Outside:
o
=
-M co e A ro
Area Properties for Various Cross Sections:
Cross Section
Rectangle
r
t
ri
h
ro
Trapezoid
r
ti
to
ri
h
ro
Hollow Circle
r
a
r
ri
+
h 2
bb g g ri
+
h ti +2to 3 ti + to
For triangle: set ti or to to 0
r
z dA
area
t
lnFHG
ro ri
IKJ
FHG IKJ to
-
ti
+
ro
ti
- h
ri
to
ln
ro ri
LNM OQP 2 r2 - b2 - r2 - a2
b
A ht h ti + to
2
e j a2 - b2
4
Basic Stress Equations
Dr. D. B. Wallace
Bending Moment in Curved Beam (Inside/Outside Stresses):
Stresses for the inside and outside fibers of a curved beam in pure bending can be approximated from the straight beam equation as modified by an appropriate curvature factor as determined from the graph below [i refers to the inside, and o refers to the outside]. The curvature factor magnitude depends on the amount of curvature (determined by the ratio r/c) and the cross section shape. r is the radius of curvature of the beam centroidal axis, and c is the distance from the centroidal axis to the inside fiber.
Centroidal Axis
c
M
r
Inside Fiber:
i =
Ki
M I
c
M
Outside Fiber:
o =
Ko
M I
c
4.0
3.5
3.0
Curvature Factor
2.5
2.0 Ki
1.5
B
b/8
Values of Ki for inside fiber as at A
B U or T
Round or Elliptical
Trapezoidal
b/6
B I or hollow rectangular
AB
b/4
A
B
b/2
B
b/3
A
B
A
b
c
A
c
b
A
c
b
A
c r
1.0
Ko
0.5
U or T
I or hollow rectangular
Round, Elliptical or Trapezoidal Values of Ko for outside fiber as atB
0
1 2 3 4 5 6 7 8 9 10 11
Amount of curvature, r/c
5
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